User blog:Rgetar/Definition of Veblen-like functions for natural superscripts and subscripts
For designations used here see User_blog:Rgetar/Definitions update. Arrays I designate arrays as ordinals, for example: 1, 2, 3 = Ω2 + Ω2 + 3 Ordinals as sums Transfinite ordinal X can be represented as sum of terms Ωαγβγ in order of decreasing γ, where Ωα ≤ X < Ωα + 1, βγ < Ωα, γ < Ωα + 1. Ω1 = Ω Ω0 = ω (Note: for ω ≤ X < Ω it is Cantor normal form). leo(X), lbeo(X), lest(X; γ), lbest(X; γ) New definition of leo(X): leo(X) is last term of this sum, that is term Ωαγβγ with γ = 0: leo(X) = Ωα0β0 = β0, even if β0 = 0 (do not confuse leo(X) with lbeo(X) - last non-zero term of this sum). New definition of lbeo(X): lbeo(X) is last non-zero term of this sum: lbeo(X) = Ωαγβγ, where γ is least ordinal such as βγ > 0. New definition of lest(X; γ): lest(X; γ) is this sum with leo(X) replaced with γ: X = ... + Ωα2β2 + Ωαβ1 + β0 lest(X; γ) = ... + Ωα2β2 + Ωαβ1 + γ New definition of lbest(X; γ): lbest(X; γ) is this sum with lbeo(X) replaced with γ. (Note: for Ω ≤ X < Ω2 these new definitions coincide with old definitions). X0 New definition of X0: the same as old definition, but now new lest is used: X0 = lest(X; 0) That is X = ... + Ωα2β2 + Ωαβ1 + β0 X0 = ... + Ωα2β2 + Ωαβ1 Cofinality cof(X) is cofinality of X, that is minimal length of increasing sequence Xn such as sup(Xn) = X. n < cof(X) Predecessor X-1 is predecessor of X, that is suc(X-1) = X, if cof(X) = 1. Veblen-like functions Here is Veblen-like function: φαβ(X), where α < β, X < Ωβ + 1, φαβ(X) < Ωα + 1. (Edit: that is Ωα is maximal cardinality of output, and Ωβ is maximal cardinality of input of the function φαβ(X)). Omitting superscript and subscript φα(X) = φαα + 1(X) φβ(X) = φ0β(X) Definition of Veblen-like functions for natural α and β φαβ(X) = φαγ(φγβ(X)), where α < γ < β (Edit: formally, if β = γ, then φαβ(X) = φαβ(φββ(X)), that is X = φββ(X), so φββ(X) is identity function). φα(0) = 1 φαβ(X) = sup(φαβ(X)n) if X > 0 (X + Y)n = X + Yn (Note: in earlier versions of this blog I formulated the above rule in more complex form: Xn = lbest(X; lbeo(X)n) if X ≥ ω). \((φ^α(X)β)n = \left\{\begin{array}{lcr} φ^α(X)βn \quad \text{if} \; cof(β) > 1\\ φ^α(X)(β-1) + φ^α(X)n \quad \text{if} \; cof(β) = 1\\ \end{array}\right. \) \(δ^α(X) = \left\{\begin{array}{lcr} 0 \qquad \qquad \qquad \, \text{if} \; leo(X) = 0\\ φ^α(X-1) + 1 \quad \text{if} \; leo(X) ≠ 0\\ \end{array}\right. \) \(φ^α(X)n = \left\{\begin{array}{lcr} φ^α(X-1)·n \quad \text{if} \; X<Ω_{α+1}, \; cof(X)=1\\ φ^α(Xn) \quad \text{if} \; cof(leo(X))>1\\ \left.\begin{array}{lcr} φ^α(lest(X^0n; δ^α(X))) \quad \text{if} \; cof(X^0)<Ω_{α+1} \; \text{or} \; n=0\\ φ^α(X^0[φ^α(X)n-1]) \quad \text{if} \; cof(X^0)=Ω_{α+1}, \; n>0\\ \end{array}\right\} \; \begin{array}{lcr} \text{if} \; X≥Ω_{α+1},\\ cof(leo(X))≤1\\ \end{array}\\ \end{array}\right. \) Note: maybe, clest instead of lest. Note: this definition is also fundamental sequence system. (Edit: for fundamental sequence system we also need rules defining standard forms of ordinals, otherwise the above rules can produce different fundamental systems for the same ordinals. So we should add that the ordinals φα(X) are given in standard form). Evaluation of cofinality cof(0) = 0 cof(X) = 1 if 0 < X < ω cof(X + Y) = cof(Y) (Note: in earlier versions of this blog I formulated the above rule in more complex form: cof(X) = cof(lbeo(X)) if X ≥ ω). \(cof(φ^α(X)β) = \left\{\begin{array}{lcr} cof(β) \quad \text{if} \; cof(β) > 1\\ cof(φ^α(X)) \quad \text{if} \; cof(β) = 1\\ \end{array}\right. \) \(cof(φ^α(X)) = \left\{\begin{array}{lcr} Ω_α \quad \text{if} \; X<Ω_{α+1}, \; cof(X)=1\\ cof(X) \quad \text{if} \; cof(leo(X))>1\\ \left.\begin{array}{lcr} cof(X^0) \quad \text{if} \; cof(X^0)<Ω_{α+1}\\ ω \quad \text{if} \; cof(X^0)=Ω_{α+1}\\ \end{array}\right\} \; \text{if} \; X≥Ω_{α+1}, \; cof(leo(X))≤1\\ \end{array}\right. \) Category:Blog posts